Decoding LDPC codes over integer residue rings

This correspondence presents a multistage decoding approach for a free Z(q)-submodule of Z(q)(N) of rank K defined by a sparse (N - K) x N parity-check matrix over Z(q) where q = p(m), p = 2 and m > 1. The proposed method involves the repeated application of belief propagation decoding to exploit the natural ring epimorphism Z(q) -> Z(p)1 : r -> Sigma(l-1)(i=0) r((i))p(i) with kernel p(l)Z(q) for each l, 1 <= l <= m, where r((i))p(i) is the p-adic expansion of r. Computer simulations for codes of rate half and moderate length on an additive white Gaussian noise (AWGN) channel with various modulation schemes show that such a decoding strategy offers an additional coding gain of between 0.07-0.1 dB over a single-stage decoding approach.